Constructing with a Pair of Compasses

1Using a pair of compasses

1. Fix the pencil so that its tip is level with the metal point.
2. Set the width by laying the compasses on your ruler: the point on 0 and the pencil on the length you need.
3. Hold the compasses by the handle at the top, and twist it between your thumb and finger. Keep your other hand out of the way.

An arc is a part of a circle. In constructions we rarely need whole circles, only short arcs where two of them cross.

Worked example 1: an equilateral triangle with sides of 6 cm

1. Draw AB 6 cm long.
2. Open your compasses to 6 cm.
3. With the point on A, draw an arc above AB.
4. With the point on B, draw an arc that crosses the first arc. Label the crossing point C.
5. Join A to C and B to C. All three sides are 6 cm, and every angle is 60°.

Worked example 2: a triangle with sides of 7 cm, 5 cm and 6 cm

1. Draw the longest side, AB = 7 cm.
2. Open your compasses to 5 cm and draw an arc from A.
3. Open them to 6 cm and draw an arc from B, crossing the first arc at C.
4. Join AC and BC.

Watch out: the arcs only meet if the two shorter sides add up to more than the longest side. With 3 cm, 4 cm and 8 cm, 3 + 4 = 7, which is less than 8, so no triangle can be made.

Worked example 3: a right angle at a point P on a line

1. With the point on P, draw two small arcs cutting the line on each side of P. Label them X and Y.
2. Open the compasses wider. From X, draw an arc above P. From Y, with the same width, draw an arc crossing it at Z.
3. Join P to Z. The angle between PZ and the line is exactly 90°.

Worked example 4: a rectangle 7 cm by 4 cm

1. Draw AB 7 cm long.
2. Construct a right angle at A and a right angle at B, as in example 3.
3. Open your compasses to 4 cm. Mark D on the line from A and C on the line from B.
4. Join D to C, and check that DC is 7 cm.

Worked example 5: a regular hexagon

1. Draw a circle with the radius you want, say 4 cm.
2. Keep the compasses at exactly 4 cm. Put the point anywhere on the circle and draw an arc that cuts the circle.
3. Move the point to where the arc cut the circle, and repeat. After six steps you are back where you started.
4. Join the six points in order.

This works because lines from the centre cut a regular hexagon into six equilateral triangles, so each side equals the radius. The animation below shows another way to find the same six points.

Try it yourself

1. To construct an equilateral triangle with sides of 5 cm, how wide do you open your compasses?
2. Can you construct a triangle with sides of 2 cm, 3 cm and 6 cm? Explain.
3. You construct a regular hexagon in a circle of radius 4 cm. What is its perimeter?
4. How can you check that an angle you constructed really is a right angle?

Answers

1. 5 cm, the length of a side.
2. No. 2 + 3 = 5, which is less than 6, so the arcs never meet.
3. Each side is 4 cm, so the perimeter is 6 × 4 = 24 cm.
4. Measure it with a protractor (it should read 90°), or fit the square corner of a book or set square into it.